BOOK OF ABSTRACTS
London-Paris Bachelier Workshop
September 15–16, 2022, Paris

Invited Talks
Gaussian Principal-Agent problems with memory
Eduardo Abi Jaber
Ecole Polytechnique

Are linear contracts still optimal? (joint with Stéphane Villeneuve.)

Option Markets in the Age of Robinhood (andbefore)Albina Danilova
London School of Economics

We consider an economy in which some agents do not continuously hedge
their position in derivative assets using the underlying assets market –i.e. we study the effects of an imbalanced derivate market. We show that,even in the presence of complete markets, the imbalance significantly alters the equilibrium price process of the underlying assets: risk premia
and volatility become stochastically time varying, hence option implied
volatility is characterized by smile and smirk patterns, momentum-likeprice dynamics arise as well as price spillovers across under- lying assets.
Moreover, the derivative imbalance generates self-fulfilling equilibria, e.g.
if the imbalance takes the form of a bet on an increase in asset volatility,
then the equilibrium volatility does increase. Finally, since our formulation is extremely general, our results also apply to segmented markets
where some investments are achievable only via financial intermediation.

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Non–regular McKean–Vlasov equations andcalibration problem in local stochastic volatility
modelsMao Fabrice Djete
Ecole Polytechnique

In order to deal with the question of the existence of a calibrated local
stochastic volatility model in finance, we investigate a class of McKean–
Vlasov equations where a minimal continuity assumption is imposed on
the coefficients. Namely, the drift coefficient and, in particular, thevolatility coefficient are not necessarily continuous in the measure variable for the Wasserstein topology. In this paper, we provide an existence
result and show an approximation by N–particle system or propagationof chaos for this type of McKean–Vlasov equations. As a direct result, weare able to deduce the existence of a calibrated local stochastic volatility
model for an appropriate choice of stochastic volatility parameters. The
associated propagation of chaos result is also proved.

Learning equilibria in mean field games
Romuald Elie
Deepmind & Université Gustave Eiffel

We will present different approaches and algorithms for learning equilibria in mean field games. In particular, we will consider frameworks
where uniqueness of Nash does not hold, and see how one can approximate alternative solution concepts, such as Correlated or Coarse correlated equilibria. Applications to animal flocking, vehicule routing as wellas connections with exploration problems in Reinforcement learning will
also be presented.

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Learning with signatures
Adeline Fermanian
Mines ParisTech

Modern applications of artificial intelligence lead to high-dimensional
multivariate temporal data that pose many challenges. Through a geometric approach to data flows, the notion of signature, a representationof a process as an infinite vector of its iterated integrals, is a promising
tool. Its properties, developed in the context of rough path theory, makeit a good candidate to play the role of features, then injected in learningalgorithms. If the definition of the signature goes back to the work ofChen (1960), its use in machine learning is recent, and many questions
remain to be explored. In this talk, we will give a quick overview of thesignature definition and properties, continue with a discussion on how
it can be used in a learning context, and conclude by focusing on one
specific problems: estimation in a linear regression setting.

Bridging socioeconomic pathways of carbon emission
and credit risk
Ying Jiao
Université Claude Bernard Lyon 1

As the world is facing global climate changes, the Intergovernmental
Panel on Climate Change (IPCC) has set the idealized carbon-neutral
scenario around 2050. In the meantime, many carbon reduction scenarios, known as Shared Socioeconomic Pathways (SSPs) have been proposed in the literature. We aim to investigates the impact of transition
risk on a firm’s low-carbon production. On the one hand, we consider
a firm that searches to optimize its emission level under the double objectives of maximizing its production profit and respecting the emission
mitigation scenarios. Solving the penalized optimization problem provides the optimal emission according to a given SSP benchmark. On
the other hand, such transitions affect the firm’s credit risk. We modelthe default time by using the structural default approach and are particularly concerned with how the adopted strategies following different
SSPs scenarios may influence the firm’s default probability. Finally we
propose possible extension of the model to a large portfolio of climate
concerned firms and discuss numerical methods to compute cumulative
losses. This is a joint work with Florian Bourgey and Emmanuel Gobet.

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Optimal control of path-dependent McKean-Vlasov
SDEs in infinite dimension
Idris Kharroubi
Sorbonne Université

We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-fieldmodels driven by stochastic PDEs. We first establish the well-posednessof the state equation, and then we prove the dynamic programming
principle (DPP) in such a general framework. The crucial law invarianceproperty of the value function V is rigorously obtained, which means that
V can be viewed as a function on the Wasserstein space of probability
measures on the set of continuous functions valued in Hilbert space. We
then define a notion of pathwise measure derivative, which extends the
Wasserstein derivative due to Lions [Cours au collège de france: Théorie
des jeux à champ moyens. Audio Conference, 2006-2012], and prove a
related functional Itô formula in the spirit of Dupire [Functional itô calculus. SSRN: 1435551, 2009] and Wu and Zhang [Viscosity solutions toparabolic master equations and McKean-Vlasov SDEs with closed-loop
controls. Ann. Appl. Probab., 30(2):936–986, 2020]. The Master Bellman equation is derived from the DPP by means of a suitable notion ofviscosity solution. We provide different formulations and simplificationsof such a Bellman equation notably in the special case when there is nodependence on the law of the control. This talk is based on a joint workwith Andrea Cosso, Fausto Gozzi, Huyên Pham and Mauro Rosestolato.

Optimality in General Propagator Models with
Alpha Signals
Eyal Neuman
Imperial College London

We consider an optimal liquidation problem with a linear transient
price impact, induced by a Volterra-type kernel, while also taking into
account a general price predicting signal. We formulate this problem
as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. By identifying the problem as
an infinite dimensional stochastic control problem, we characterize thevalue function in terms of a solution to an L2 -valued backward stochastic differential equations and an to an operator-valued Riccati equation,and derive an explicit optimal trading strategy. Our results also includes
5the case of a singular power-law price impact kernel. This is a joint workwith Eduardo Abi-Jaber.

Periodic portfolio selection (with quasi-hyperbolic
discounting)Alex Tse
University College London

I will present a continuous-time portfolio selection problem faced by anagent with S-shaped preference who maximizes the utilities derived fromthe portfolio’s periodic performance over an infinite horizon. I will firstoutline the solution method under a baseline setup. Then I will introduce
a time-inconsistent version of the problem featuring quasi-hyperbolic
discounting. If the agent is sophisticated who seeks a consistent planning
strategy, the problem can then be analyzed via a static mean field gamewhere theoretical characterization of the optimal strategy is provided.
This talk is based on two different joint works with Harry Zheng andYushi Hamaguchi respectively.

Neural networks for high-dimensional parametric
option pricing: Expression rates and exposure
evaluations
Linus Wunderlich
Queen Mary University London

In this talk we will discuss the deep parametric PDE method for parametric option pricing in high dimensions, underlying theoretical results
for neural networks and an application in risk management.
The deep parametric PDE method uses deep neural networks to solve
parametric partial differential equations, such as those arising in option
pricing. Especially for a large number of risk factors, the efficiency ofneural networks for high dimensional problems is beneficial. We investigate this efficiency theoretically by proving new approximation rates for
networks with smooth activation functions.
Finally an application to exposure calculations numerically confirms
the efficiency of the method. We compute expected exposures and potential future exposures for up to 10 underlying assets and compare theperformance to a competing sparse grid method.

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Exploration-exploitation trade-off forcontinuous-time reinforcement learning
Yufei Zhang
London School of Economics

Recently, reinforcement learning (RL) has attracted substantial researchinterests. Much of the attention and success, however, has been for thediscrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored andwith limited progress. In particular, characterising sample efficiency forcontinuous-time RL algorithms remains a challenging and open problem.
In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise
exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learningalgorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities forcorrelated continuous-time observations, and applying stochastic control
theory to quantify the performance gap between applying greedy policies
derived from estimated and true models.

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Contributed Talks
A Theoretical Analysis of Guyon’s Volatility Model
Ofelia Bonesini
Univ. Padova

We provide a thorough analysis of the path-dependent volatility model
introduced by Guyon, proving existence and uniqueness of a strong solution, characterising its behaviour at boundary points, providing asymptotic closed-form option prices as well as deriving small-time behaviour
estimates. This is a joint work with Doctor Chloé Lacombe and Professor Antoine Jacquier.

Diffusive limit approximation of pure jump optimalergodic control problems
Lorenzo Croissant
CEREMADE, Université Paris Dauphine-PSL and Criteo AI Lab

Motivated by the design of fast reinforcement learning algorithms in
the context of high-frequency ad-auctions, we study the diffusive limitof a class of pure-jump ergodic stochastic control problems. We show
that, whenever the intensity of jumps is large enough, the approximation
error is governed by the Hölder continuity of the Hessian matrix of thesolution to the limit ergodic partial differential equation. Using thelimit approximation can lead to significant reduction in the numericalresolution cost.

8Complexity Reduction for Exposure Measurementsof Financial Derivatives
Domagoj Demeterfi
Queen Mary University London

The key element in quantifying counterparty credit risk, namely, estimation of exposure measures, requires many, often expensive, evaluationsof the pricing functions. This talk explores the potential of complexity
reduction techniques in exposure measurements. The work is motivatedby the fact that the pricing functions repeatedly evaluated in the calculations are typically sufficiently regular and so admit accurate approximations by simpler functions. We investigate the implications of complexity
reduction in a developed theoretical framework. We show that an error
in any exposure measurement induced by complexity reduction is not
greater than an error of the underlying pricing function approximation
measured in the uniform norm. On the other hand, we show that asmall Lp norm of the function approximation error does not guarantee
an equally small error for a general exposure measure. We consider the
Chebyshev interpolation as a particular instance of the approximation
method used for a complexity reduction, given its appealing theoreticaland practical properties for applications in finance. Conducted numerical experiments covering several financial derivatives in different option
pricing setups confirm the effectiveness of the approach. Obtained theoretical and practical results suggest great potential benefits and uncoverpossible pitfalls of the complexity reduction approach to exposure measurements.

Equilibrium Asset Pricing with stochastic investment
opportunities and proportional transactional costs
Justin Gwee
London School of Economics and Political Science

In view of the increasing automatisation of financial markets, correctly
specifying the tradeoff between frequent trading and transaction costs
becomes paramount. In this talk, we consider the impact of proportional transaction costs on risk-sharing equilibria between two agentswith mean-variance preferences. The state of the economy is modelledby an underlying diffusion process leading to stochastic investment opportunities for the agents. Here, our model allows us to observe new
effects resulting from the interplay between the transaction costs and9the stochastic investment opportunities. This includes solving a twodimensional free boundary problem similar to those arising from singular stochastic control problems and generalises earlier results by Gonon,
Muhle-Karbe and Shi (2021) that have been established in a model withconstant coefficients. The talk is based on joint work with Christoph
Czichowsky and Mihalis Zervos.

Linear Quadratic Principal Multi-Agent Incentive
Problems with Applications to Development ofRenewable Energy
Annika Kemper
Center for Mathematical Economics at Bielefeld University (Germany)In this work we provide a class of tractable Principal Multi-Agent incentives problems. We follow the resolution methodology of Cvitanić et al.
(2018) in an extended linear-quadratic setting with exponential utilities
and a multi-dimensional state process of Ornstein-Uhlenbeck type. Our
analysis results in a closed-form second-best contract being of rebate
form. Moreover, we investigate the optimal regulation of energy production reflecting the long-term goals of the Paris Climate Agreement. In
particular, we provide optimal regulatory incentives in a monopolistic
and competitive Two-Agent setting to encourage green investments andlimit carbon emissions, while simultaneously stabilizing the total energy
production. A numerical study highlights the impact of the designed
second-best contract in comparison to business as usual. In particular,
we illustrate how to encourage renewable investments, how to avoid carbon emissions, how to ensure a stable production, and how competition
affects incentives.

10An Optimal stopping Mean Field Game with
common noise model for energy transition underscenario uncertainty.
Marcos Leutscher
CREST, ENSAE, Institut Polytechnique de Paris

We propose a discrete time mean-field game model for the industry dynamics in the electricity market. The market is subject to common noisein the form of random shocks in the carbon price. These shocks depend
on a macroeconomic scenario which is not observed by the agents, but
can be partially deduced from the frequency of the shocks. Due to thispartial observation feature, the common noise is non-Markovian. We
consider two classes of agents: conventional producers and renewableproducers. The former choose an optimal moment to exit the market and the latter choose an optimal moment to enter the market bybuilding a renewable plant. The agents interact through the market
price which is determined by a merit order mechanism, involving an exogenous stochastic demand process. We study the existence of Nashequilibria in the resulting mean-field game problem of optimal stoppingwith common noise, developing a novel linear programming approach forthese problems. We illustrate our model by an example inspired in the
UK electricity market. Joint work with Roxana Dumitrescu and Peter
Tankov.

Impulse Control of a Linear Diffusion: an Explicitly
Solvable Problem
Zhesheng Liu
London School of Economics

We consider a stochastic impulse control problem that is motivated byseveral applications in areas such as the optimal cashflow management or
the optimal exploitation of a natural resource. In particular, we consider
a stochastic system whose uncontrolled state dynamics are modelled by anon-explosive positive linear diffusion. The control that can be applied tothis system takes the form of one-sided impulsive action. The objectiveof the control problem is to maximise a discounted performance criterion
that rewards the effect of control action but involves a fixed cost at eachtime of a control intervention. We derive the complete solution to thisproblem under general assumptions. It turns out that the solution cantake four qualitatively different forms. In two of the four cases, there
exist only ε-optimal control strategies.

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A Stochastic Target Problem for Branching
Diffusions
Antonio OcelloLPSM – Sorbonne Université

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition
for which a control allows the underlying branching process to reach atarget set at a finite terminal time for each of its branches. This problem
is motivated by an example from fintech where we look for the superreplication price of options on blockchain based cryptocurrencies. We
first state a dynamic programming principle for the value function of
the stochastic target problem. We then show that the value function
can be reduced to a new function with a finite dimensional argument bya so called branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational
inequality.

Fast Calibration using Complex-Step Sobolev
Training
Bouazza Saadeddine
LaMME, Univ. Evry / Crédit Agricole CIB

We present a new fast calibration technique where we propose to train
neural networks to directly perform the orthogonal projection of simulated payoffs of the calibration instrument with randomized model parameters and we enrich the learning task by including path-wise sensitivities of the payoffs with respect to model and product parameters.
We show that this particular instance of Sobolev training can be reformulated in a way that requires computing only (stochastic) directional
derivatives and we provide a fast, memory-efficient and numerically stable approach to compute those using complex-step differentiation. Our
experiment with a fixed-grid piece-wise linear local volatility example
demonstrates that one can get competitive price approximations without having to train the neural network on Monte Carlo prices and thatboth data-set construction and training can be done in reasonable time
while preserving a very general framework that can be applied to a broad
range of pricing models.

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Diffusion of carbon price in a credit portfolio
through macroeconomic factors
Lionel Sopgoui
LPSM, Univ. Paris Cité ; Imperial College London ; Groupe BPCE

We study how the introduction of a carbon price in a closed economy
propagates in a credit portfolio and precisely describe how carbon pricedynamics affects credit risk variables (or metrics) such as probability ofdefault, expected and unexpected loss. We adapt a classical monetarymodel (Monetary policy, inflation, and the business cycle: an introduction to the new Keynesian framework and its applications, Galì, 2015)
to take into account two carbon taxes (one on firms’ production and one
on households’ consumption) (Climate-related scenarios for financial stability assessment: an application to France, Allen et al., 2020), whichyields a closed-form relationship between carbon taxes – calibrated onrealised GHG emissions and on carbon prices – and real GDP growth.
We then introduce carbon price dynamics in a stochastic multi-sectoralmodel, where taxes are calibrated of sectoral households and firms’ greenhouse gases emissions. This yields the sensitivity of firms’ production
and households’ consumption of each sector to carbon price and the relationships between sectors. This allows us to analyse the short-term
effects (quarterly or annually) of carbon taxes as opposed to standard
Integrated Assessment Models (such as REMIND (Description of the
REMIND model (Version 1.6), Luderer et al., 2015)), which are not
only deterministic but only capture long-term trends of climate transition policy. Finally, we use a discount cash flow methodology (Problems
with the discounted cash flow method: discrepancy between theory andpractice, Seijdell, 2015) to compute firms’ values and use them in the
Merton model to describe the deformations of the credit risk metrics
with a carbon price. We observe that, on the one hand, the introductionof a carbon price not only increases banking fees charged to clients (materialised by the provisions via the expected loss) but will also reducebanks’ profitability (via the economic capital that is calculated from theunexpected loss). In addition the added randomness of our model adds
extra flexibility to compute confidence intervals for a more accurate risk
assessment. We also compute the sensitivities of the credit risk metrics
with respect to changes in the carbon price, yielding further criteria fora more accurate assessment of climate risk in a credit portfolio. This
work provides a preliminary methodology to calculate the evolution of
credit risk metrics of a multi-sectorial credit portfolio, starting from a
climate transition scenario described by a carbon price.

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Convergence rate of RBSDE by penalisation method
and its Monte-Carlo approximation
Wanqing Wang
Ecole PolytechniqueIn this paper, we study the numerical solution (Ŷ , Ẑ, K̂) for Reflected
Backward Stochastic Equations (RBSDEs in short) by the penalisation
approach and we apply this on the pricing problem of American option.
Considering the barrier as a generalised Itô process (which is natural forCall and Put), we first show that the penalisation term can converge to 0
at order 1 or 21 with respect to the constant of penalty and then we deducethat the solution of penalised BSDE can converge to the one of RBSDE
at respectively the same order as the penalisation term. Followed by
the results in the continuous case, when the generator does not dependon Z, the error analysis is also established for the discrete implicite
scheme of PBSDE and in the end, using the Monte-Carlo method, weshow numerically that the error decrease faster than the order we have
proved.

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